Nonlinear Control Sy.. - Free

A.7. CHAPTER 10 335
Proof of Theorem 10.3: To prove the theorem, we proceed as follows.
The single vector g is clearly involutive and thus, the Frobenius theorem guarantees
that for each xp E D there exist a neighborhood SZ of xo and n - 1 linearly independent
smooth functions µl, . ,µ._1(x) such that
Lyµi(x) = 0 for 1 < i < n - 1 dx E SZ
also, by (A.73) in Lemma A.2, Vh(x), , VLf-2h(x) are linearly independent. Thus,
defining
T(x) =
r µ1(x) 1
i.e., with h(x), , Lf lh(x) in the last r rows of T, we have that
VLFlh(xo) Vspan{Vµ1, ...,
= rank (15 (xo)J I = n.
which implies that (xo) # 0. Thus, T is a diffeomorphism in a neighborhood of xo, and
the theorem is proved.
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